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Hubble's law at current time is as follows. $$v=H(t_0)r$$

But if you look at the explanation on Wikipedia:

Strictly speaking, neither $v$ nor $D$ in the formula are directly observable, because they are properties now of a galaxy, whereas our observations refer to the galaxy in the past, at the time that the light we currently see left it.

For relatively nearby galaxies (redshift $z$ much less than unity), $v$ and $D$ will not have changed much, and $v$ can be estimated using the formula $v=zc$ where $c$ is the speed of light. This gives the empirical relation found by Hubble.

For distant galaxies, $v$ (or $D$) cannot be calculated from $z$ without specifying a detailed model for how $H$ changes with time. The redshift is not even directly related to the recession velocity at the time the light set out, but it does have a simple interpretation: ($1+z$) is the factor by which the universe has expanded while the photon was travelling towards the observer.

My question is as follows:

Current proper distance indicates where the galaxy should be by expansion when photons from the past start and reach us now.

In the expression of Hubble's law, Hubble's law has a current proper distance. Therefore, Hubble's law at current time already has a meaning that implies information about photons in the past. It's strange to say that Hubble's law only holds close distance because of the inability to observe current information.

Why does $cz=Hd$ exist only at a close distance?

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  • $\begingroup$ Short answer: cosmic expansion is not constant, but H = H(t). $\endgroup$ Sep 22 '21 at 8:58
  • $\begingroup$ @planetmaker Of course, I know that the Hubble parameter changes over time. “Strictly speaking, neither 𝑣( nor 𝐷) in the formula are directly observable, because they are properties now of a galaxy, whereas our observations refer to the galaxy in the past, at the time that the light we currently see left it.” The meaning of this phrase doesn't make sense. Do means past information by definition. But it represents the current information. $\endgroup$ Sep 22 '21 at 10:11
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I'm not sure I understand the question, but I'll try to provide some context for the paragraph from Wikipedia.

It's ambiguous what law Hubble should be credited with finding, because there are many different notions of distance that can be used in cosmology, and all of them satisfy some equation of the form $H_0 D = cz + O(z^2)$, and Hubble's original data points all had $z^2<0.000015$, and they were noisy enough to be equally consistent with any of these notions of distance.

There is one version of "Hubble's law" that is exactly correct at all distances and times in any (exactly homogeneous) FLRW cosmology, regardless of the specific parameters of the model. That is $v=HD$ where $D$ is the metric distance measured at a constant cosmological time and $v$ is the derivative of $D$ with respect to cosmological time.

However, the relationship between the quantities in that Hubble law and directly measurable quantities, such as $z$, luminosity, and angular size, does depend on the details of the model. There is no law as simple and universal as $v=HD$ that relates measurable quantities.

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